Yukawa-SYK model and self-tuned quantum criticality

Non-Fermi liquids (NFLs) are a class of strongly interacting gapless fermionic systems without long-lived quasiparticle excitations. An important group of NFL models feature itinerant fermions coupled to soft bosonic fluctuations near a quantum-critical point and are widely believed to capture the essential physics of many unconventional superconductors. However, numerically, the direct observation of a canonical NFL behavior in such systems, characterized by a power-law form in the Green's function, has been elusive. Here, we consider a Sachdev-Ye-Kitaev (SYK)-like model with random Yukawa interaction between critical bosons and fermions (dubbed the Yukawa-SYK model). We show that it is immune from the minus-sign problem and hence can be solved exactly via large-scale quantum Monte Carlo simulation beyond the large-$N$ limit accessible to analytical approaches. Our simulation demonstrates that the Yukawa-SYK model features “self-tuned quantum criticality”; namely, the system is critical independent of the bosonic bare mass. We put these results to the test at finite $N$, and our unbiased numerics reveal clear evidence of these exotic quantum-critical NFL properties—the power-law behavior in the Green's function of fermions and bosons—which propels the theoretical understanding of critical Planckian metals and unconventional superconductors.

Figure 1

Yukawa-SYK model. There are $M$ quantum dots labeled by $\{i,j\}$, and each dot has $N$ flavors labeled by $\{\alpha ,\beta \}$. Bosons are given by antisymmetric fields ${\varphi }_{\alpha \beta }$. Fermions are coupled to bosons through a random Yukawa coupling $t_{i\alpha ,j\beta }$.

Figure 2

Theoretical result of (a) $G_f$ and (b) $G_b$ at $N=4M\to \infty , {\omega }_0=1, m_0=2$, and various temperatures. Here, we show them in log-log plots. The auxiliary dashed lines, whose slopes are $1-x$ and $2x$, respectively, show that $G_f(\tau ,0)\propto {\left(\frac{\pi }{\beta sin(\pi \tau /\beta )}\right)}^{1-x}$ and $G_b(\tau ,0)\propto {\left(\frac{\pi }{\beta sin(\pi \tau /\beta )}\right)}^{2x}$ at $\tau \to \frac{\beta }{2}$, when $\beta$ is large enough.

Figure 3

Inverse transition temperature ${\beta }_c$ from NFL to superconductivity as a function of the ratio ${\omega }_0/m_0$ for $N=4M$ and $N=M$, obtained from solving Eq. (10) at large $N$.

Figure 4

QMC results at $N=4M, m_0=2, {\omega }_0=1$, and $\beta =16$ for $M=2,3,4$. (a) Green's function of fermions $G_f(\tau ,0)$ versus $\tau$ in the range of $\tau \in [0,\beta ]$. Blue, red, and yellow dots are DQMC data, and the black dashed line is the large-$N$ result. (b) Green's function of bosons $G_b(\tau ,0)$ versus $\tau$ in the range of $\tau \in [0,\beta ]$. (c) and (d) The same as above, but in a special log-log scale as in Fig. 2. The convergence towards the large-$N$ results as $(M,N)$ increase is obvious. In (a)–(d), the error bars denote the variation of the Green's function for different disorder realizations. The progressively smaller error bars as $M$ increases indicate that the randomness of the coupling self-averages.

Figure 5

(a) and (c) show $G_b$ and $G_f$ of 20 different disorder realizations with $M=N=9, \beta =24, m_0=2$, and ${\omega }_0=1$. The Green's functions are very close to each other. (b) and (d) present the difference between theoretical (large $N$) results $G^{\mathrm{Th}}(\beta /2,0)$ and QMC numerical simulation data $G(\beta /2,0)$. It is clear that as $M\left(N\right)$ increases, the distance between QMC and analytics gradually reduces. The relative standard deviations in the QMC data are also decreasing. The parameters are set at $M=N, \beta =24, m_0=2$, and ${\omega }_0=1$.

Figure 6

QMC results of (a) $G_b$ and (b) $G_f$ at $M=4,N=16, m_0={\omega }_0=1$, and $\beta =16,20,24$. In log-log plots, we see that as $\beta$ increases, large-$N$ results are basically unchanged, while curves of the QMC simulation progressively deviate from the large-$N$ value due to increasing finite-$N$ corrections.

Figure 7

Self-tuned quantum criticality with different boson masses in log-log plots. (a) $G_b(\tau ,0)$ from a free-boson model with $m_0=1$ and $\beta =16$. The exponential decay in imaginary time is evident with $ln\left[G_b(\tau =\beta /2)\right]\sim -7$. (b) and (c) show the $G_b(\tau ,0)$ from the Yukawa-SYK model in Eq. (1) with different mass $m_0={\omega }_0$ (b) and $m_0=2{\omega }_0$ (c) with ${\omega }_0=1$ at $M=4,N=16$ and $\beta =16$. Blue circles are DQMC data, and the red dashed lines are the large-$N$ result. Power-law decay of $G_b$ at low temperatures and long-time limit is clear to see in the log-log plots. These results reveal the self-tuned quantum criticality in our system.

Figure 8

DQMC results of Matsubara Green's function $G_b$ at $M=N, m_0={\omega }_0=1$, and $\beta =16$. We plot the zero-frequency component and its error bars. Glassy behaviors are seen.

Figure 9

A Feynman diagram involving replica-off-diagonal processes that are not suppressed by $1/MN$ in the model given by Eq. (C1). The two fermion loops carry different replicon indices.